What is a triangle made of. Types of triangles: right-angled, acute-angled, obtuse-angled

Triangle - definition and general concepts

A triangle is such a simple polygon, consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.

All vertices of any triangle, regardless of its variety, are indicated by capital Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, only not in capital letters, but in small letters. So, for example, a triangle with vertices labeled A, B, and C has sides a, b, c.

If we consider a triangle in Euclidean space, then this is such geometric figure, which was formed using three segments connecting three points that do not lie on one straight line.

Look closely at the picture above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms corners inside it.

Types of triangles

According to the size, angles of triangles, they are divided into such varieties as: Rectangular;
Acute-angled;
obtuse.

Right-angled triangles are triangles that have one right angle and the other two have acute angles.

Acute-angled triangles are those in which all of its angles are acute.

And if a triangle has one obtuse angle, and the other two angles are acute, then such a triangle belongs to obtuse angles.

Each of you is well aware that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:

Isosceles;
Equilateral;
Versatile.

Task: Draw different types triangles. Give them a definition. What difference do you see between them?

Basic properties of triangles

Although these simple polygons may differ from each other in the size of the angles or sides, but in each triangle there are basic properties that are characteristic of this figure.

In any triangle:

The sum of all its angles is 180º.
If it belongs to equilateral, then each of its angles is equal to 60º.
An equilateral triangle has identical and equal angles to each other.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are located equal angles, and vice versa.
If we take a triangle and extend its side, then in the end we will form an external angle. It is equal to the sum of the interior angles.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:

1.a< b + c, a >b-c;
2.b< a + c, b >a-c;
3.c< a + b, c >a-b.

Exercise

The table shows the already known two angles of the triangle. Knowing the total sum of all the angles, find what the third angle of the triangle is equal to and enter in the table:

1. How many degrees does the third angle have?
2. What kind of triangles does it belong to?

Equivalence Triangles

I sign

II sign

III sign

Height, bisector and median of a triangle

The height of a triangle - the perpendicular drawn from the top of the figure to its opposite side, is called the height of the triangle. All heights of a triangle intersect at one point. The intersection point of all 3 altitudes of a triangle is its orthocenter.

A segment drawn from a given vertex and connecting it in the middle of the opposite side is the median. Medians, like the heights of a triangle, have one common point intersection, the so-called center of gravity of the triangle or centroid.

The bisector of a triangle is a segment that connects the vertex of an angle and a point on the opposite side, and also divides this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.

The segment that connects the midpoints of the 2 sides of the triangle is called the midline.

Historical reference

Such a figure as a triangle was known in ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron's formula, the study of the property of a triangle switched to more high level, but still, it happened more than two thousand years ago.

In the XV - XVI centuries began to conduct a lot of research on the properties of the triangle, and as a result, such a science as planimetry arose, which was called the "New Triangle Geometry".

A scientist from Russia N. I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application both in mathematics and in physics and cybernetics.

Thanks to the knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when compiling maps, measuring areas, and even when designing various mechanisms.

What is the most famous triangle? This is, of course, the Bermuda Triangle! It got its name in the 50s because of geographical location points (vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The peaks of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.

Assignment: What theories about the Bermuda Triangle have you heard?

Do you know that in Lobachevsky's theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemannian geometry, the sum of all the angles of a triangle is greater than 180º, while in Euclid's writings it is equal to 180 degrees.

Homework

Solve a crossword puzzle on a given topic

Crossword questions:

1. What is the name of the perpendicular drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the larger one from the sides of the triangle?
6. Name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90 °?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, capital letter And is...?
11. What is the name of the segment that divides the angle of the triangle in half.

Questions about triangles:

1. Give a definition.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called wonderful.
7. What instrument can measure the angle?
8. If the hands of the clock show 21 hours. What angle do the hour hands form?
9. At what angle does a person turn if he is given the command "to the left", "around"?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?

Standard notation

Triangle with vertices A, B And C denoted as (see Fig.). The triangle has three sides:

The lengths of the sides of a triangle are indicated by lowercase Latin letters (a, b, c):

The triangle has the following angles:

The angles at the corresponding vertices are traditionally denoted by Greek letters (α, β, γ).

Signs of equality of triangles

A triangle on the Euclidean plane can be uniquely (up to congruence) defined by the following triplets of basic elements:

1. a, b, γ (equality on two sides and the angle lying between them);
2. a, β, γ (equality in side and two adjacent angles);
3. a, b, c (equality on three sides).

Signs of equality of right triangles:

1. along the leg and hypotenuse;
2. on two legs;
3. along the leg and acute angle;
4. hypotenuse and acute angle.

Some points in the triangle are "paired". For example, there are two points from which all sides are visible either at an angle of 60° or at an angle of 120°. They're called dots Torricelli. There are also two points whose projections on the sides lie at the vertices right triangle. This - points of Apollonius. Points and such as are called Brocard points.

Direct

In any triangle, the center of gravity, the orthocenter and the center of the circumscribed circle lie on the same straight line, called Euler line.

The line passing through the center of the circumscribed circle and the Lemoine point is called Brokar's axis. Apollonius points lie on it. The Torricelli points and the Lemoine point also lie on the same straight line. The bases of the outer bisectors of the angles of a triangle lie on the same straight line, called axis of external bisectors. The points of intersection of the lines containing the sides of the orthotriangle with the lines containing the sides of the triangle also lie on the same line. This line is called orthocentric axis, it is perpendicular to the Euler line.

If we take a point on the circumscribed circle of a triangle, then its projections on the sides of the triangle will lie on one straight line, called Simson's straight line given point. Simson's lines of diametrically opposite points are perpendicular.

triangles

• A triangle with vertices at the bases of the cevians drawn through given point, is called cevian triangle this point.
• A triangle with vertices in the projections of a given point onto the sides is called under the skin or pedal triangle this point.
• A triangle with vertices at the second intersection points of lines drawn through the vertices and a given point, with a circumscribed circle, is called cevian triangle. A cevian triangle is similar to a subdermal one.

circles

• Inscribed circle is a circle tangent to all three sides of the triangle. She is the only one. The center of the inscribed circle is called incenter.
• Circumscribed circle- a circle passing through all three vertices of the triangle. The circumscribed circle is also unique.
• Excircle- a circle tangent to one side of a triangle and the extension of the other two sides. There are three such circles in a triangle. Their radical center is the center of the inscribed circle of the median triangle, called Spieker's point.

The midpoints of the three sides of a triangle, the bases of its three altitudes, and the midpoints of the three line segments connecting its vertices to the orthocenter lie on a single circle called circle of nine points or Euler circle. The center of the nine-point circle lies on the Euler line. A circle of nine points touches an inscribed circle and three excircles. The point of contact between an inscribed circle and a circle of nine points is called Feuerbach point. If from each vertex we lay out triangles on straight lines containing sides, orthoses equal in length to opposite sides, then the resulting six points lie on one circle - Conway circles. In any triangle, three circles can be inscribed in such a way that each of them touches two sides of the triangle and two other circles. Such circles are called Malfatti circles. The centers of the circumscribed circles of the six triangles into which the triangle is divided by medians lie on one circle, which is called Lamun circle.

A triangle has three circles that touch two sides of the triangle and the circumscribed circle. Such circles are called semi-inscribed or Verrier circles. The segments connecting the points of contact of the Verrier circles with the circumscribed circle intersect at one point, called Verrier point. It serves as the center of the homothety, which takes the circumscribed circle to the incircle. The points of tangency of the Verrier circles with the sides lie on a straight line that passes through the center of the inscribed circle.

The line segments connecting the tangent points of the inscribed circle with the vertices intersect at one point, called Gergonne point, and the segments connecting the vertices with the points of contact of the excircles - in Nagel point.

Ellipses, parabolas and hyperbolas

Inscribed conic (ellipse) and its perspective

An infinite number of conics (ellipses, parabolas, or hyperbolas) can be inscribed in a triangle. If we inscribe an arbitrary conic in a triangle and connect the points of contact with opposite vertices, then the resulting lines will intersect at one point, called perspective conics. For any point of the plane that does not lie on a side or on its extension, there exists an inscribed conic with a perspective at that point.

Steiner's ellipse circumscribed and cevians passing through its foci

An ellipse can be inscribed in a triangle that touches the sides at the midpoints. Such an ellipse is called Steiner inscribed ellipse(its perspective will be the centroid of the triangle). The described ellipse, which is tangent to lines passing through vertices parallel to the sides, is called circumscribed by the Steiner ellipse. If an affine transformation ("skew") translates the triangle into a regular one, then its inscribed and circumscribed Steiner ellipse will go into an inscribed and circumscribed circle. Cevians drawn through the foci of the described Steiner ellipse (Skutin points) are equal (Skutin's theorem). Of all the described ellipses, the described Steiner ellipse has the smallest area, and of all the inscribed largest area has an inscribed Steiner ellipse.

Brocard's ellipse and its perspector - Lemoine point

An ellipse with foci at Brokar's points is called Brocard ellipse. Its perspective is the Lemoine point.

Properties of an inscribed parabola

Kiepert parabola

The perspectives of the inscribed parabolas lie on the circumscribed Steiner ellipse. The focus of an inscribed parabola lies on the circumscribed circle, and the directrix passes through the orthocenter. A parabola inscribed in a triangle whose directrix is ​​the Euler line is called Kiepert's parabola. Its perspective is the fourth point of intersection of the circumscribed circle and the circumscribed Steiner ellipse, called Steiner point.

Cypert's hyperbole

If the described hyperbola passes through the intersection point of the heights, then it is equilateral (that is, its asymptotes are perpendicular). The intersection point of the asymptotes of an equilateral hyperbola lies on a circle of nine points.

Transformations

If the lines passing through the vertices and some point not lying on the sides and their extensions are reflected with respect to the corresponding bisectors, then their images will also intersect at one point, which is called isogonally conjugate the original one (if the point lay on the circumscribed circle, then the resulting lines will be parallel). Many pairs of remarkable points are isogonally conjugate: the center of the circumscribed circle and the orthocenter, the centroid and the Lemoine point, the Brocard points. The Apollonius points are isogonally conjugate to the Torricelli points, and the center of the incircle is isogonally conjugate to itself. Under the action of isogonal conjugation, straight lines go into circumscribed conics, and circumscribed conics into straight lines. Thus, the Kiepert hyperbola and the Brocard axis, the Enzhabek hyperbola and the Euler line, the Feuerbach hyperbola and the line of centers of the inscribed circle are isogonally conjugate. The circumscribed circles of subdermal triangles of isogonally conjugate points coincide. The foci of the inscribed ellipses are isogonally conjugate.

If, instead of a symmetric cevian, we take a cevian whose base is as far from the middle of the side as the base of the original one, then such cevians will also intersect at one point. The resulting transformation is called isotomic conjugation. It also maps lines to circumscribed conics. The Gergonne and Nagel points are isotomically conjugate. Under affine transformations, isotomically conjugate points pass into isotomically conjugate ones. At isotomy conjugation, the described Steiner ellipse passes into the straight line at infinity.

If, in the segments cut off by the sides of the triangle from the circumscribed circle, circles are inscribed that touch the sides at the bases of the cevians drawn through a certain point, and then the points of contact of these circles are connected to the circumscribed circle with opposite vertices, then such lines will intersect at one point. The transformation of the plane, which compares the resulting point to the starting point, is called isocircular transformation. The composition of the isogonal and isotomic conjugations is the composition of the isocircular transformation with itself. This composition is a projective transformation that leaves the sides of the triangle in place, and translates the axis of the outer bisectors into a straight line at infinity.

If we continue the sides of the cevian triangle of some point and take their intersection points with the corresponding sides, then the resulting intersection points will lie on one straight line, called trilinear polar starting point. Orthocentric axis - trilinear polar of the orthocenter; the trilinear polar of the center of the inscribed circle is the axis of the outer bisectors. The trilinear polars of the points lying on the circumscribed conic intersect at one point (for the circumscribed circle this is the Lemoine point, for the circumscribed Steiner ellipse it is the centroid). The composition of the isogonal (or isotomic) conjugation and the trilinear polar is a duality transformation (if the point isogonally (isotomically) conjugate to the point lies on the trilinear polar of the point , then the trilinear polar of the point isogonally (isotomically) conjugate to the point lies on the trilinear polar of the point ).

Cubes

Relationships in a triangle

Note: in this section, , , are the lengths of the three sides of the triangle, and , , are the angles lying respectively opposite these three sides (opposite angles).

triangle inequality

In a non-degenerate triangle, the sum of the lengths of its two sides is greater than the length of the third side, in a degenerate one it is equal. In other words, the lengths of the sides of a triangle are related by the following inequalities:

The triangle inequality is one of the axioms of metrics.

Triangle sum of angles theorem

Sine theorem

where R is the radius of the circle circumscribed around the triangle. It follows from the theorem that if a< b < c, то α < β < γ.

Cosine theorem

Tangent theorem

Other ratios

Metric ratios in a triangle are given for:

Solving Triangles

The calculation of unknown sides and angles of a triangle, based on known ones, has historically been called "triangle solutions". In this case, the above general trigonometric theorems are used.

Area of ​​a triangle

The following inequalities hold for the area:

Calculating the area of ​​a triangle in space using vectors

Let the vertices of the triangle be at the points , , .

Let's introduce the area vector . The length of this vector is equal to the area of ​​the triangle, and it is directed along the normal to the plane of the triangle:

Let , where , , are the projections of the triangle onto the coordinate planes. Wherein

and likewise

The area of ​​the triangle is .

An alternative is to calculate the lengths of the sides (using the Pythagorean theorem) and then using the Heron formula.

Triangle theorems

Desargues theorem: if two triangles are perspective (the lines passing through the corresponding vertices of the triangles intersect at one point), then their respective sides intersect on one straight line.

Sond's theorem: if two triangles are perspective and orthologous (perpendiculars dropped from the vertices of one triangle to the sides opposite to the corresponding vertices of the triangle, and vice versa), then both orthology centers (points of intersection of these perpendiculars) and the perspective center lie on one straight line perpendicular to the perspective axis (straight line from the Desargues theorem).

Tasks:

1. Introduce students to different types of triangles depending on the type of angles (rectangular, acute-angled, obtuse-angled). Learn to find triangles and their types in the drawings. To fix the basic geometric concepts and their properties: straight line, segment, ray, angle.

2. Development of thinking, imagination, mathematical speech.

3. Education of attention, activity.

During the classes

I. Organizational moment.

How much do we need guys?
For our skillful hands?
Draw two squares
And they have a big circle.
And then some more circles
Triangle cap.
So it came out very, very
Cheerful Weird.

II. Announcement of the topic of the lesson.

Today in the lesson we will make a trip around the city of Geometry and visit the Triangles microdistrict (that is, we will get acquainted with different types of triangles depending on their angles, we will learn to find these triangles in the drawings.) We will conduct a lesson in the form of a “competition game” by commands.

1 team - “Segment”.

2 team - "Ray".

Team 3 - "Corner".

And the guests will represent the jury.

The jury will guide us along the way

And will not leave without attention. (Evaluate by points 5,4,3,...).

And on what will we travel around the city of Geometry? Remember what types of passenger transport are in the city? There are so many of us, which one shall we choose? (Bus).

Bus. Clearly, briefly. Boarding begins.

Let's get comfortable and start our journey. Team captains get tickets.

But these tickets are not easy, and the tickets are “tasks”.

III. Repetition of the material covered.

First stop"Repeat."

Question for all teams.

Find a straight line in the drawing and name its properties.

Without end and edge, the line is straight!
At least a hundred years go along it,
You won't find the end of the road!

• The straight line has neither beginning nor end - it is infinite, so it cannot be measured.

Let's start our competition.

Protecting your team names.

(All teams read the first questions and discuss. In turn, the team captains read out the questions, 1 team reads 1 question).

1. Show a segment in the drawing. What is called a cut. Name its properties.

• The part of a straight line bounded by two points is called a line segment. A line segment has a beginning and an end, so it can be measured with a ruler.

(Team 2 reads 1 question).

1. Show the beam on the drawing. What is called a beam. Name its properties.

• If you mark a point and draw a part of a straight line from it, you get an image of a beam. The point from which a part of the line is drawn is called the beginning of the ray.

The beam has no end, so it cannot be measured.

(Team 3 reads 1 question).

1. Show the angle on the drawing. What is called an angle. Name its properties.

• Drawing two rays from one point, a geometric figure is obtained, which is called an angle. An angle has a vertex, and the rays themselves are called sides of the angle. Angles are measured in degrees using a protractor.

Fizkultminutka (to the music).

IV. Preparing to study new material.

Second stop"Fabulous".

On a walk, the Pencil met different angles. I wanted to say hello to them, but I forgot the name of each of them. Pencil will have to help.

(The angles of the study are checked using the model of a right angle).

Assignment to teams. Read questions #2 and discuss.

Team 1 reads question 2.

2. Find a right angle, give a definition.

• An angle of 90° is called a right angle.

Team 2 reads question 2.

2. Find an acute angle, give a definition.

• An angle less than a right angle is called an acute angle.

Team 3 reads question 2.

2. Find an obtuse angle, give a definition.

An angle greater than a right angle is called obtuse.

In the microdistrict where Pencil liked to walk, all the corners differed from other residents in that the three of us always walked, the three of us drank tea, and the three of us went to the cinema. And the Pencil could not understand what kind of geometric figure three angles together make up?

A poem will give you a hint.

You on me, you on him
Look at all of us.
We have everything, we have everything
We only have three!

Which shape is being referred to?

• About the triangle.

What shape is called a triangle?

• A triangle is a geometric figure that has three vertices, three angles, and three sides.

(Learners show a triangle in the drawing, name the vertices, angles and sides).

Vertices: A, B, C (points)

Angles: BAC, ABC, BCA.

Sides: AB, BC, CA (segments).

V. Physical education:

stomp your foot 8 times,
Clap your hands 9 times
we will squat 10 times,
and bend over 6 times
we'll jump straight
so many (triangle display)
Hey, yes, count! Game and more!

VI. Learning new material.

Soon the corners became friends and became inseparable.

And now we will call the microdistrict: the Triangles microdistrict.

The third stop is “Znayka”.

What are the names of these triangles?

Let's give them names. And let's try to formulate the definition ourselves.

Team 3 answers.

1 team will find and show obtuse triangles.

2 command will find and show right triangles.

3 command will find and show acute triangles.

VIII. The next stop is Thinking.

Assignment to all teams.

After shifting 6 sticks, make 4 equal triangles from the lantern.

What kind of angles are triangles? (Acute-angled).

IX. Summary of the lesson.

What neighborhood did we visit?

What types of triangles are you familiar with?

The science of geometry tells us what a triangle, square, cube is. IN modern world it is studied in schools by all without exception. Also, a science that directly studies what a triangle is and what properties it has is trigonometry. She explores in detail all the phenomena associated with data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems related to them.

What is a triangle? Definition

This is a flat polygon. It has three corners, which is clear from its name. It also has three sides and three vertices, the first of which are segments, the second are points. Knowing what two angles are equal to, you can find the third one by subtracting the sum of the first two from the number 180.

What are triangles?

They can be classified according to various criteria.

First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first have acute angles, that is, those that are less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.

Right triangle

Impossible not to talk about what right triangle.

Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The other two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. With its help, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can recall the isosceles. This is one in which two of the sides are equal, and two of the angles are also equal.

What is the leg and hypotenuse?

The leg is one of the sides of a triangle that form an angle of 90 degrees. The hypotenuse is the remaining side that is opposite right angle. From it, a perpendicular can be lowered onto the leg. The ratio of the adjacent leg to the hypotenuse is called the cosine, and the opposite is called the sine.

- what are its features?

It is rectangular. Its legs are three and four, and the hypotenuse is five. If you saw that the legs of this triangle are equal to three and four, you can be sure that the hypotenuse will be equal to five. Also, according to this principle, it can be easily determined that the leg will be equal to three if the second is equal to four, and the hypotenuse is five. To prove this statement, you can apply the Pythagorean theorem. If two legs are 3 and 4, then 9 + 16 = 25, the root of 25 is 5, that is, the hypotenuse is 5. Also Egyptian triangle called rectangular, the sides of which are 6, 8 and 10; 9, 12 and 15 and other numbers with a ratio of 3:4:5.

What else could be a triangle?

Triangles can also be inscribed and circumscribed. The figure around which the circle is described is called inscribed, all its vertices are points lying on the circle. A circumscribed triangle is one in which a circle is inscribed. All its sides are in contact with it at certain points.

How is

The area of ​​any figure is measured in square units (square meters, square millimeters, square centimeters, square decimeters, etc.). This value can be calculated in a variety of ways, depending on the type of triangle. The area of ​​any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite angle, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle between these sides, and divide this by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this, you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the four values ​​obtained. Next, find out the number that came out. The area of ​​an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by which is circumscribed around it times four.

The area of ​​the described triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: we square the side, multiply the resulting figure by the root of three, then divide this number by four. Similarly, you can calculate the height of a triangle in which all sides are equal, for this you need to multiply one of them by the root of three, and then divide this number by two.

Triangle theorems

The main theorems that are associated with this figure are the Pythagorean theorem, described above, and cosines. The second (sine) is that if you divide any side by the sine of the angle opposite to it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosine) is that if the sum of the squares of the two sides is subtracted from their product, multiplied by two and the cosine of the angle located between them, then the square of the third side will be obtained.

Dali triangle - what is it?

Many, faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. The Dali Triangle is common name three places that are closely related to life famous artist. Its “tops” are the house where Salvador Dali lived, the castle that he gave to his wife, and the museum of surrealistic paintings. During the tour of these places you can learn a lot. interesting facts about this peculiar creative artist known all over the world.